Ad hoc and Sensor Networks Chapter 9: Localization & positioning

Transcription

1 Ad hoc and Sensor Networks Chapter 9: Localization & positioning Holger Karl Computer Networks Group Universität Paderborn

2 Goals of this chapter Means for a node to determine its physical position (with respect to some coordinate system) or symbolic location Using the help of Anchor nodes that know their position Directly adjacent Over multiple hops Using different means to determine distances/angles locally SS 05 Ad hoc & sensor networs - Ch 9: Localization & positioning 2

3 Overview Basic approaches Trilateration Multihop schemes SS 05 Ad hoc & sensor networs - Ch 9: Localization & positioning 3

4 Localization & positioning Determine physical position or logical location Coordinate system or symbolic reference Absolute or relative coordinates Options Centralized or distributed computation Scale (indoors, outdoors, global, ) Sources of information Metrics Accuracy (how close is an estimated position to the real position?) Precision (for repeated position determinations, how often is a given accuracy achieved?) Costs, energy consumption, SS 05 Ad hoc & sensor networs - Ch 9: Localization & positioning 4

5 Main approaches (information sources) Proximity Exploit finite range of wireless communication E.g.: easy to determine location in a room with infrared room number announcements (Tri-/Multi-)lateration and angulation Use distance or angle estimates, simple geometry to compute position estimates Scene analysis Radio environment has characteristic signatures Can be measured beforehand, stored, compared with current situation (x = 2, y = 1) Angle φ 1 (x = 5, y = 4) r 2 r 1 Length known r 3 (x = 8, y = 2) Angle φ 2 SS 05 Ad hoc & sensor networs - Ch 9: Localization & positioning 5

6 Estimating distances RSSI Received Signal Strength Indicator Send out signal of known strength, use received signal strength and path loss coefficient to estimate distance Problem: Highly error-prone process Shown: PDF for a fixed RSSI PDF PDF Distance Distance Signal strength SS 05 Ad hoc & sensor networs - Ch 9: Localization & positioning 6

7 Estimating distances other means Time of arrival (ToA) Use time of transmission, propagation speed, time of arrival to compute distance Problem: Exact time synchronization Time Difference of Arrival (TDoA) Use two different signals with different propagation speeds Example: ultrasound and radio signal Propagation time of radio negligible compared to ultrasound Compute difference between arrival times to compute distance Problem: Calibration, expensive/energy-intensive hardware SS 05 Ad hoc & sensor networs - Ch 9: Localization & positioning 7

8 Determining angles Directional antennas On the node Mechanically rotating or electrically steerable On several access points Rotating at different offsets Time between beacons allows to compute angles φ β α γ 2φ 3φ SS 05 Ad hoc & sensor networs - Ch 9: Localization & positioning 8

9 Some range-free, single-hop localization techniques Overlapping connectivity: Position is estimated in the center of area where circles from which signal is heard/not heard overlap Approximate point in triangle Determine triangles of anchor nodes where node is inside, overlap them A C B G E F Check whether inside a given triangle move node or simulate movement by asking neighbors D Only approximately correct?? SS 05 Ad hoc & sensor networs - Ch 9: Localization & positioning 9

10 Overview Basic approaches Trilateration Multihop schemes SS 05 Ad hoc & sensor networs - Ch 9: Localization & positioning 10

11 Trilateration Assuming distances to three points with known location are exactly given Solve system of equations (Pythagoras!) (x i,y i ) : coordinates of anchor point i, r i distance to anchor i (x u, y u ) : unknown coordinates of node Subtracting eq. 3 from 1 & 2: Rearranging terms gives a linear equation in (x u, y u )! SS 05 Ad hoc & sensor networs - Ch 9: Localization & positioning 11

12 Trilateration as matrix equation Rewriting as a matrix equation: Example: (x 1, y 1 ) = (2,1), (x 2, y 2 ) = (5,4), (x 3, y 3 ) = (8,2), r 1 = , r 2 = 2, r 3 = 3!(x u,y u ) = (5,2) SS 05 Ad hoc & sensor networs - Ch 9: Localization & positioning 12

13 Trilateration with distance errors What if only distance estimation r i0 = r i + ε i available? Use multiple anchors, overdetermined system of equations Use (x u, y u ) that minimize mean square error, i.e, SS 05 Ad hoc & sensor networs - Ch 9: Localization & positioning 13

14 Minimize mean square error Look at square of the of Euclidean norm expression (note that for all vectors v) Look at derivative with respect to x, set it equal to 0: Normal equation Has unique solution (if A has full rank), which gives desired minimal mean square error Essentially similar for angulation as well SS 05 Ad hoc & sensor networs - Ch 9: Localization & positioning 14

15 Overview Basic approaches Trilateration Multihop schemes SS 05 Ad hoc & sensor networs - Ch 9: Localization & positioning 15

16 Multihop range estimation How to estimate range to a node to which no direct radio communication exists? B No RSSI, TDoA, X But: Multihop communication is possible A Idea 1: Count number of hops, assume length of one hop is known (DV-Hop) Start by counting hops between anchors, divide known distance Idea 2: If range estimates between neighbors exist, use them to improve total length of route estimation in previous method (DV-Distance) C SS 05 Ad hoc & sensor networs - Ch 9: Localization & positioning 16

17 Iterative multilateration Assume some nodes can hear at least three anchors (to perform triangulation), but not all I: (18,20) II: A B (?,?) (?,?) (2,10) (2,10) (18,20) A B (12,14) (?,?) Idea: let more and more nodes compute position estimates, spread position knowledge in the network Problem: Errors accumulate III: (8,0) (2,10) (8,0) C (?,?) (18,20) A B (12,14) C (?,?) (38,5) (30,12) (38,5) IV: (8,0) (2,10) (8,0) C (?,?) (18,20) A B (12,14) C (22,2) (38,5) (30,12) (38,5) SS 05 Ad hoc & sensor networs - Ch 9: Localization & positioning 17

18 Probabilistic position description Similar idea to previous one, but accept problem that position of nodes is only probabilistically known Represent this probability explicitly, use it to compute probabilities for further nodes SS 05 Ad hoc & sensor networs - Ch 9: Localization & positioning 18

19 Conclusions Determining location or position is a vitally important function in WSN, but fraught with many errors and shortcomings Range estimates often not sufficiently accurate Many anchors are needed for acceptable results Anchors might need external position sources (GPS) Multilateration problematic (convergence, accuracy) SS 05 Ad hoc & sensor networs - Ch 9: Localization & positioning 19